Extremal Problems in Disjoint Cycles
and Graph Saturation
Elyse Yeager
yeager2@illinois.edu
University of Alaska, Fairbanks
09 February 2015
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Theory
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Theory
trade agreement
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
2 / 32
Introduction to Graph Theory
trade agreement
no trade agreement
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Theory
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
2 / 32
Introduction to Graph Theory
direct contact
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Theory
direct contact
no direct contact
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Theory
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
Turán Problem
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
Turán Problem
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
Turán Problem
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
Turán Problem
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
3 / 32
Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
Turán Problem
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
3 / 32
Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
Turán Problem
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Extremal Graph Theory
Extremal Graph Theory
Find maximum and minimum values of a graph parameter, given criteria.
Turán Problem
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Graph Saturation
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Saturation
Turán: Maximum number of edges in a graph with no forbidden subgraph.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Saturation
Turán: Maximum number of edges in a graph with no forbidden subgraph.
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Saturation
Turán: Maximum number of edges in a graph with no forbidden subgraph.
Extra property:
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
5 / 32
Introduction to Graph Saturation
Turán: Maximum number of edges in a graph with no forbidden subgraph.
Extra property:
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
5 / 32
Introduction to Graph Saturation
Turán: Maximum number of edges in a graph with no forbidden subgraph.
Extra property:
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
5 / 32
Introduction to Graph Saturation
Turán: Maximum number of edges in a graph with no forbidden subgraph.
Extra property: adding any edge results in a forbidden subgraph
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Introduction to Graph Saturation
Turán: Maximum number of edges in a graph with no forbidden subgraph.
Extra property: adding any edge results in a forbidden subgraph
Forbidden:
Graph Saturation
A graph G is H-saturated if G contains no forbidden H subgraph, but
adding any edge to G gives rise to an H subgraph.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Saturation Number
Easy Question
What is the minimum number of edges in a graph G with no forbidden
subgraph H?
Forbidden:
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Saturation Number
Easy Question
What is the minimum number of edges in a graph G with no forbidden
subgraph H?
Interesting Question: Saturation Number (Erdős-Hajnal-Moon)
What is the minimum number of edges in a graph G that is H-saturated?
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Saturation Number
Easy Question
What is the minimum number of edges in a graph G with no forbidden
subgraph H?
Interesting Question: Saturation Number (Erdős-Hajnal-Moon)
What is the minimum number of edges in a graph G that is H-saturated?
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Saturation Number
Easy Question
What is the minimum number of edges in a graph G with no forbidden
subgraph H?
Interesting Question: Saturation Number (Erdős-Hajnal-Moon)
What is the minimum number of edges in a graph G that is H-saturated?
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Saturation Number
Easy Question
What is the minimum number of edges in a graph G with no forbidden
subgraph H?
Interesting Question: Saturation Number (Erdős-Hajnal-Moon)
What is the minimum number of edges in a graph G that is H-saturated?
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Edge-Colored Graph Saturation
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
Forbidden in Blue
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
(i)
E. Yeager ( yeager2@illinois.edu )
Forbidden in Blue
Extremal Problems
09 February 2015
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Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
(ii)
E. Yeager ( yeager2@illinois.edu )
Forbidden in Blue
Extremal Problems
09 February 2015
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Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
(ii)
E. Yeager ( yeager2@illinois.edu )
Forbidden in Blue
Extremal Problems
09 February 2015
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Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
(ii)
E. Yeager ( yeager2@illinois.edu )
Forbidden in Blue
Extremal Problems
09 February 2015
8 / 32
Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
(ii)
E. Yeager ( yeager2@illinois.edu )
Forbidden in Blue
Extremal Problems
09 February 2015
8 / 32
Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
(ii)
E. Yeager ( yeager2@illinois.edu )
Forbidden in Blue
Extremal Problems
09 February 2015
8 / 32
Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Forbidden in Red
Forbidden in Blue
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
8 / 32
Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Ferrara-Kim-Y, 2014
Iterated Recoloring: method for using results about (uncolored) graph
saturation to illuminate problems in edge-colored graph saturation.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
8 / 32
Edge-Colored Graph Saturation
Edge-Colored Saturation
A graph G is sautrated with respect to forbidden subgraphs H1 , . . . , Hk if:
(i) there exists a k-coloring with no Hi in color i, and
(ii) G is edge-maximal with respect to this property
Ferrara-Kim-Y, 2014
Iterated Recoloring: method for using results about (uncolored) graph
saturation to illuminate problems in edge-colored graph saturation.
Future Research
Hanson-Toft Conjecture: determine edge-colored saturation number
for complete graphs
Broader application of Iterated Recoloring
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
7
7
7
E. Yeager ( yeager2@illinois.edu )
7
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
7
7
Induced
7
E. Yeager ( yeager2@illinois.edu )
7
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
7
7
7
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
7
7
Not Induced
7
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
10 / 32
Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
10 / 32
Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
7
7
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
10 / 32
Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
10 / 32
Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
7
7
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
10 / 32
Induced Saturation
Induced Subgraph
A graph H is an induced subgraph of a graph G if H can be obtained from
G by deleting vertices.
7
7
E. Yeager ( yeager2@illinois.edu )
Induced
Extremal Problems
09 February 2015
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Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
77
7
7 7
E. Yeager ( yeager2@illinois.edu )
Forbidden
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
77
7
7 7
E. Yeager ( yeager2@illinois.edu )
Forbidden
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
7
7
7
77
E. Yeager ( yeager2@illinois.edu )
Forbidden
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
11 / 32
Induced Saturation
Idea for Induced Saturation:
Adding or deleting any edge produces a forbidden induced subgraph.
Forbidden
Warning: not defined for all forbidden subgraphs!
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Martin-Smith
Definition that works for all forbidden subgraphs. (technical)
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Induced Saturation
Martin-Smith
Definition that works for all forbidden subgraphs. (technical)
Behrens-Erbes-Santana-Yager-Y, 2015+
A large number of common families fit into the simpler definition
paw
E. Yeager ( yeager2@illinois.edu )
odd cycles
stars
Extremal Problems
matchings
09 February 2015
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Induced Saturation
Martin-Smith
Definition that works for all forbidden subgraphs. (technical)
Behrens-Erbes-Santana-Yager-Y, 2015+
A large number of common families fit into the simpler definition
Using simpler definition, minimize number of edges
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
12 / 32
Induced Saturation
Martin-Smith
Definition that works for all forbidden subgraphs. (technical)
Behrens-Erbes-Santana-Yager-Y, 2015+
A large number of common families fit into the simpler definition
Using simpler definition, minimize number of edges
Future Research
Characterize when simple definition suffices
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
12 / 32
Induced Saturation
Martin-Smith
Definition that works for all forbidden subgraphs. (technical)
Behrens-Erbes-Santana-Yager-Y, 2015+
A large number of common families fit into the simpler definition
Using simpler definition, minimize number of edges
Future Research
Characterize when simple definition suffices
Determine “minimum number of edges” for specific graphs, families
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
12 / 32
Induced Saturation
Martin-Smith
Definition that works for all forbidden subgraphs. (technical)
Behrens-Erbes-Santana-Yager-Y, 2015+
A large number of common families fit into the simpler definition
Using simpler definition, minimize number of edges
Future Research
Characterize when simple definition suffices
Determine “minimum number of edges” for specific graphs, families
Martin-Smith Definition: find examples of non-monotonicity
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
12 / 32
Cycles
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Cycles
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Cycles
Cycle
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
14 / 32
Cycles
Cycle
E. Yeager ( yeager2@illinois.edu )
Forest: acyclic graph
Tree: connected acyclic graph
Extremal Problems
09 February 2015
14 / 32
Cycles
Cycle
E. Yeager ( yeager2@illinois.edu )
Forest: acyclic graph
Tree: connected acyclic graph
Extremal Problems
09 February 2015
14 / 32
Cycles
Cycle
E. Yeager ( yeager2@illinois.edu )
Forest: acyclic graph
Tree: connected acyclic graph
Extremal Problems
09 February 2015
14 / 32
Cycles
Cycle
E. Yeager ( yeager2@illinois.edu )
Forest: acyclic graph
Tree: connected acyclic graph
Extremal Problems
09 February 2015
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Cycles
Cycle
Forest: acyclic graph
Tree: connected acyclic graph
Degree of a vertex, d(v ): number of edges attached to v .
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Cycles
Cycle
Forest: acyclic graph
Tree: connected acyclic graph
Degree of a vertex, d(v ): number of edges attached to v .
Minimum degree of a graph G : δ(G )
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
15 / 32
Minimum Degree of a Forest
Fact
Any graph with minimum degree at least 2 contains a cycle. That is, every
forest has minimum degree 1 or 0.
Proof : Suppose a graph has minimum degree at least 2.
A cycle exists.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Disjoint Cycles and Corrádi-Hajnal
Corrádi-Hajnal, 1963
If G is a graph on n vertices with n ≥ 3k and δ(G ) ≥ 2k, then G contains
k disjoint cycles.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Disjoint Cycles and Corrádi-Hajnal
Corrádi-Hajnal, 1963
If G is a graph on n vertices with n ≥ 3k and δ(G ) ≥ 2k, then G contains
k disjoint cycles.
Sharpness:
k
k
k
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
16 / 32
Disjoint Cycles and Corrádi-Hajnal
Corrádi-Hajnal, 1963
If G is a graph on n vertices with n ≥ 3k and δ(G ) ≥ 2k, then G contains
k disjoint cycles.
Sharpness:
k
k
k
E. Yeager ( yeager2@illinois.edu )
2k − 1
Extremal Problems
09 February 2015
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Enomoto, Wang
Corrádi-Hajnal, 1963
If G is a graph on n vertices with n ≥ 3k and δ(G ) ≥ 2k, then G contains
k disjoint cycles.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Enomoto, Wang
Corrádi-Hajnal, 1963
If G is a graph on n vertices with n ≥ 3k and δ(G ) ≥ 2k, then G contains
k disjoint cycles.
σ2 (G ) := min{d(x) + d(y ) : xy 6∈ E (G )}
That is: low-degree vertices are all connected;
other vertices have higher degree to compensate
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Enomoto, Wang
Corrádi-Hajnal, 1963
If G is a graph on n vertices with n ≥ 3k and δ(G ) ≥ 2k, then G contains
k disjoint cycles.
σ2 (G ) := min{d(x) + d(y ) : xy 6∈ E (G )}
That is: low-degree vertices are all connected;
other vertices have higher degree to compensate
Enomoto 1998, Wang 1999
If G is a graph on n vertices with n ≥ 3k and σ2 (G ) ≥ 4k − 1, then G
contains k disjoint cycles.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Enomoto, Wang
Corrádi-Hajnal, 1963
If G is a graph on n vertices with n ≥ 3k and δ(G ) ≥ 2k, then G contains
k disjoint cycles.
σ2 (G ) := min{d(x) + d(y ) : xy 6∈ E (G )}
That is: low-degree vertices are all connected;
other vertices have higher degree to compensate
Enomoto 1998, Wang 1999
If G is a graph on n vertices with n ≥ 3k and σ2 (G ) ≥ 4k − 1, then G
contains k disjoint cycles.
Implies Corrádi-Hajnal
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Extremal Problems
09 February 2015
17 / 32
Enomoto, Wang
Enomoto 1998, Wang 1999
If G is a graph on n vertices with n ≥ 3k and σ2 (G ) ≥ 4k − 1, then G
contains k disjoint cycles.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
18 / 32
Enomoto, Wang
Enomoto 1998, Wang 1999
If G is a graph on n vertices with n ≥ 3k and σ2 (G ) ≥ 4k − 1, then G
contains k disjoint cycles.
Sharpness:
k
k
k
E. Yeager ( yeager2@illinois.edu )
2k − 1
Extremal Problems
09 February 2015
18 / 32
Enomoto, Wang
Enomoto 1998, Wang 1999
If G is a graph on n vertices with n ≥ 3k and σ2 (G ) ≥ 4k − 1, then G
contains k disjoint cycles.
Sharpness:
k
k
k
2k − 1
α(G ) > n − 2k
n = 3k
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Extremal Problems
09 February 2015
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Kierstead-Kostochka-Y, 2014+
Enomoto 1998, Wang 1999
If G is a graph on n vertices with n ≥ 3k and σ2 (G ) ≥ 4k − 1, then G
contains k disjoint cycles.
k
k
k
2k − 1
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
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Extremal Problems
09 February 2015
19 / 32
Kierstead-Kostochka-Y, 2014+
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
20 / 32
Kierstead-Kostochka-Y, 2014+
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
n ≥ 3k + 1
k
k
k
E. Yeager ( yeager2@illinois.edu )
2k − 1
Extremal Problems
k
09 February 2015
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Kierstead-Kostochka-Y, 2014+
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
k = 1:
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Extremal Problems
09 February 2015
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Kierstead-Kostochka-Y, 2014+
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
k = 2:
u
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
v
20 / 32
Kierstead-Kostochka-Y, 2014+
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
k = 3:
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Extremal Problems
09 February 2015
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Kierstead-Kostochka-Y, 2014+
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
σ2 = 4k − 4:
k−3
2r
k+1
K2t
k+3
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
2r − 2
09 February 2015
20 / 32
Dirac’s Question
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Extremal Problems
09 February 2015
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Dirac: (2k − 1)-connected without k disjoint cycles
Dirac, 1963
What (2k − 1)-connected graphs do not have k disjoint cycles?
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
22 / 32
Dirac: (2k − 1)-connected without k disjoint cycles
Dirac, 1963
What (2k − 1)-connected graphs do not have k disjoint cycles?
Observation:
G is (2k − 1) connected
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Extremal Problems
09 February 2015
22 / 32
Dirac: (2k − 1)-connected without k disjoint cycles
Dirac, 1963
What (2k − 1)-connected graphs do not have k disjoint cycles?
Observation:
G is (2k − 1) connected ⇒ δ(G ) ≥ 2k − 1
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
22 / 32
Dirac: (2k − 1)-connected without k disjoint cycles
Dirac, 1963
What (2k − 1)-connected graphs do not have k disjoint cycles?
Observation:
G is (2k − 1) connected ⇒ δ(G ) ≥ 2k − 1 ⇒ σ2 (G ) ≥ 4k − 2
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
22 / 32
Dirac: (2k − 1)-connected without k disjoint cycles
Dirac, 1963
What (2k − 1)-connected graphs do not have k disjoint cycles?
Observation:
G is (2k − 1) connected ⇒ δ(G ) ≥ 2k − 1 ⇒ σ2 (G ) ≥ 4k − 2
KKMY: Holds for σ2 (G ) ≥ 4k − 3
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
22 / 32
Dirac: (2k − 1)-connected without k disjoint cycles
Dirac, 1963
What (2k − 1)-connected graphs do not have k disjoint cycles?
Answer to Dirac’s Question for Simple Graphs
Let k ≥ 2. Every graph G with (i) |G | ≥ 3k and (ii) δ(G ) ≥ 2k − 1
contains k disjoint cycles if and only if
if k is odd and |G | = 3k, then G 6= 2Kk ∨ Kk , and
α(G ) ≤ |G | − 2k, and
if k = 2 then G is not a wheel.
k
k
k
E. Yeager ( yeager2@illinois.edu )
2k − 1
Extremal Problems
09 February 2015
22 / 32
Dirac: (2k − 1)-connected without k disjoint cycles
Dirac, 1963
What (2k − 1)-connected graphs do not have k disjoint cycles?
Answer to Dirac’s Question for Simple Graphs
Let k ≥ 2. Every graph G with (i) |G | ≥ 3k and (ii) δ(G ) ≥ 2k − 1
contains k disjoint cycles if and only if
if k is odd and |G | = 3k, then G 6= 2Kk ∨ Kk , and
α(G ) ≤ |G | − 2k, and
if k = 2 then G is not a wheel.
Further:
characterization for multigraphs
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Extremal Problems
09 February 2015
22 / 32
Multigraphs
Simple Graph
Smallest cycle: 3 vertices
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Multigraph
Smallest cycle: 1 vertex
Extremal Problems
09 February 2015
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(2k − 1)-connected multigraphs with no k disjoint cycles
Answer to Dirac’s Question for multigraphs: Kierstead-Kostochka-Yeager
Combinatorica, to appear
Let k ≥ 2 and n ≥ k. Let G be an n-vertex graph with simple degree at least 2k − 1
and no loops. Let F be the simple graph induced by the strong edgs of G , α0 = α0 (F ),
and k 0 = k − α0 . Then G does not contain k disjoint cycles if and only if one of the
following holds:
n + α0 < 3k;
|F | = 2α0 (i.e., F has a perfect matching) and either (i) k 0 is odd and
G − F = Yk 0 ,k 0 , or (ii) k 0 = 2 < k and G − F is a wheel with 5 spokes;
G is extremal and either (i) some big set is not incident to any strong edge, or (ii)
for some two distinct big sets Ij and Ij 0 , all strong edges intersecting Ij ∪ Ij 0 have a
common vertex outside of Ij ∪ Ij 0 ;
n = 2α0 + 3k 0 , k 0 is odd, and F has a superstar S = {v0 , . . . , vs } with center v0
such that either (i) G − (F − S + v0 ) = Yk 0 +1,k 0 , or (ii) s = 2, v1 v2 ∈ E (G ),
G − F = Yk 0 −1,k 0 and G has no edges between {v1 , v2 } and the set X0 in G − F ;
k = 2 and G is a wheel, where some spokes could be strong edges;
k 0 = 2, |F | = 2α0 + 1 = n − 5, and G − F = C5 .
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Extremal Problems
09 February 2015
24 / 32
k 0 odd, F has a perfect matching
Example: k = 8, α0 = 3, k 0 = 5.
k0
k0
k0
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Extremal Problems
09 February 2015
25 / 32
Big independent set, incident to no multiple edges
2k − 1
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Extremal Problems
09 February 2015
26 / 32
Equitable Coloring
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Extremal Problems
09 February 2015
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Equitable Coloring
Definition
An equitable k-coloring of a graph G is a partition of its vertices into
independent sets (called color classes) such that any two color classes
differ in size by at most one.
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Extremal Problems
09 February 2015
28 / 32
Equitable Coloring
Definition
An equitable k-coloring of a graph G is a partition of its vertices into
independent sets (called color classes) such that any two color classes
differ in size by at most one.
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Extremal Problems
09 February 2015
28 / 32
Equitable Coloring
Definition
An equitable k-coloring of a graph G is a partition of its vertices into
independent sets (called color classes) such that any two color classes
differ in size by at most one.
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Extremal Problems
09 February 2015
28 / 32
Equitable Coloring and Cycles
n = 3k
If G has n = 3k vertices, then G has an equitable k-coloring if and only if
G has k disjoint cycles (all triangles).
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
28 / 32
Equitable Coloring and Cycles
n = 3k
If G has n = 3k vertices, then G has an equitable k-coloring if and only if
G has k disjoint cycles (all triangles).
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
28 / 32
Ore Conditions
Chen-Lih-Wu Conjecture
If χ(G ), ∆(G ) ≤ k and Kk,k 6⊆ G , then G is equitably k-colorable.
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Extremal Problems
09 February 2015
29 / 32
Ore Conditions
Chen-Lih-Wu Conjecture
If χ(G ), ∆(G ) ≤ k and Kk,k 6⊆ G , then G is equitably k-colorable.
Kierstead-Kostochka-Molla-Y, 2014+
If G is a k-colorable 3k-vertex graph such that for each edge xy ,
d(x) + d(y ) ≤ 2k + 1, then G is equitably k-colorable, or is one of several
exceptions.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
29 / 32
Ore Conditions
Chen-Lih-Wu Conjecture
If χ(G ), ∆(G ) ≤ k and Kk,k 6⊆ G , then G is equitably k-colorable.
Kierstead-Kostochka-Molla-Y, 2014+
If G is a k-colorable 3k-vertex graph such that for each edge xy ,
d(x) + d(y ) ≤ 2k + 1, then G is equitably k-colorable, or is one of several
exceptions.
Equivalent
If G is a graph on 3k vertices with σ2 (G ) ≥ 4k − 3, then G contains k
disjoint cycles, or is one of several exceptions, or G is not k-colorable.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Ore Conditions
Kierstead-Kostochka-Molla-Y, 2014+
If G is a k-colorable 3k-vertex graph such that for each edge xy ,
d(x) + d(y ) ≤ 2k + 1, then G is equitably k-colorable, or is one of several
exceptions.
Equivalent
If G is a graph on 3k vertices with σ2 (G ) ≥ 4k − 3, then G contains k
disjoint cycles, or is one of several exceptions, or G is not k-colorable.
KKY, 2014+
For k ≥ 4, if G is a graph on n vertices with n ≥ 3k + 1 and
σ2 (G ) ≥ 4k − 3, then G contains k disjoint cycles if and only if
α(G ) ≤ n − 2k.
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Exceptions
k=3
Equitable coloring:
Cycles:
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Extremal Problems
09 February 2015
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Exceptions
Equitable coloring:
c
Kk
2k − c
Cycles:
k
k
k
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Extremal Problems
09 February 2015
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Exceptions
Equitable coloring:
Kk−1
2k
Cycles:
K2k
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k−1
Extremal Problems
09 February 2015
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Future Directions
Chorded Cycles
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Extremal Problems
09 February 2015
31 / 32
Future Directions
Chorded Cycles
Dirac-Erdős
(number of high-degree vertices)−(number of low-degree vertices)
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Extremal Problems
09 February 2015
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Future Directions
Chorded Cycles
Dirac-Erdős
(number of high-degree vertices)−(number of low-degree vertices)
Proven: quadratic
E. Yeager ( yeager2@illinois.edu )
Extremal Problems
09 February 2015
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Future Directions
Chorded Cycles
Dirac-Erdős
(number of high-degree vertices)−(number of low-degree vertices)
Proven: quadratic
Perhaps linear?
2k − 1
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Extremal Problems
09 February 2015
31 / 32
Thanks for Listening!
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Extremal Problems
09 February 2015
32 / 32